Journal of Statistical Physics

, Volume 123, Issue 3, pp 601–614 | Cite as

Irreducible Free Energy Expansion and Overlaps Locking in Mean Field Spin Glasses



Following the works of Guerra, 1995; Aizenmar and Contucci, J. State. Phys. 92 (5–6): 765–783 (1998), we introduce a diagrammatic formulation for a cavity field expansion around the critical temperature. This approach allows us to obtain a theory for the overlap's fluctuations and, in particular, the linear part of the Ghirlanda–Guerra relationships (GG) (often called Aizenman–Contucci polynomials (AC)) in a very simple way. We show moreover how these constraints are “superimposed” by the symmetry of the model with respect to the restriction required by thermodynamic stability. Within this framework it is possible to expand the free energy in terms of these irreducible overlaps fluctuations and in a form that simply put in evidence how the complexity of the solution is related to the complexity of the entropy.

Key Words

Cavity field Ghirlanda–Guerra stochastic stability Aizenman-Contucci polynomials 


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  1. 1.
    D. Sherrington and S. Kirkpatrick, A solvable model of spin glass. Phys. Rew. Lett. 35:1792–1796 (1975).CrossRefADSGoogle Scholar
  2. 2.
    D. Sherrington and S. Kirkpatrick, Infinite ranged models of spin glass. Phys. Rew. B17:4384–4403 (1978).ADSGoogle Scholar
  3. 3.
    M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond World Scientific Publishing (1987).Google Scholar
  4. 4.
    L. A. Pasteur and M. V. Shcherbina, The absence of self averaging of the order parameter in SK model. J. Stat. Phys. 62:1–19 (1991).CrossRefGoogle Scholar
  5. 5.
    S. Ghirlanda and F. Guerra, General properties of overlap probability distributions in disordered spin systems. J. Phys. A 31:9149–9155 (1998).MATHMathSciNetCrossRefADSGoogle Scholar
  6. 6.
    F. Guerra, Sum rules for the free energy in the mean field spin glass model. Field Institute Comm. 30 (2001).Google Scholar
  7. 7.
    E. Marinari, G. Parisi, J. Ruiz-Lorenzo, and F. Ritort, Numerical evidence for spontaneously broken replica symmetry in 3D spin glasses. arXiv: cond-mat/950836v1.Google Scholar
  8. 8.
    G. Parisi, On the probabilistic formulation of the replica approach to spin glasses. arXiv: cond-mat/9801081v1Google Scholar
  9. 9.
    F. Guerra, The cavity method in the mean field spin glass model. Functional representation of the thermodynamic variables. Advances in dynamical system and quantum physics. World Scientific (1995).Google Scholar
  10. 10.
    F. Guerra, About the cavity fields in mean field spin glass models. arXiv: cond-mat/0307673v1.Google Scholar
  11. 11.
    M. Aizenman and P. Contucci, On the stability of the quenched state in mean field spin glass models. J. Stat. Phys. 92(5–6):765–783 (1998).Google Scholar
  12. 12.
    G. Parisi, Stochastic stability Disordered and complex system. AIP Conf. Proc.Google Scholar
  13. 13.
    M. Talagrand, Spin Glasses: A Challenge for Mathematicians. Springer-Verlag, Berlin (2003).Google Scholar
  14. 14.
    F. Guerra and F. L. Toninelli, The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230(1):71–79 (2002).Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsKing's College LondonLondonUK
  2. 2.Dipartimento di FisicaUniversita' di Roma “La Sapienza,”RomaItaly

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